// Make newform 287.2.a.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_287_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_287_2_a_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_287_2_a_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then poly := [-3, 6, 4, -6, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-3, 0, 1, 0, 0], [4, -1, -6, 0, 1], [-2, -3, 5, 1, -1]]; Rf_basisdens := [1, 1, 1, 1, 1]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_287_a();" function MakeCharacter_287_a() N := 287; order := 1; char_gens := [206, 211]; v := [1, 1]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; function MakeCharacter_287_a_Hecke(Kf) return MakeCharacter_287_a(); end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 2; raw_aps := [[0, -1, 0, 0, 0], [1, -1, 0, 0, 0], [-1, 0, -1, 1, 0], [1, 0, 0, 0, 0], [-1, 1, -2, 0, 1], [1, 0, 1, 0, 1], [3, 0, 0, -1, -2], [-1, 1, -1, -2, -1], [1, 1, 0, 2, 0], [0, 1, 1, 1, -1], [3, 2, 1, -1, 0], [-2, 1, 1, -2, 0], [-1, 0, 0, 0, 0], [-1, 0, 0, -3, 0], [0, 3, 1, -2, 2], [3, -2, 1, 3, 0], [3, -4, 3, 1, 2], [5, -1, 0, 2, 1], [-1, -2, -1, 1, 2], [-5, -1, 0, -4, -3], [9, -3, 0, 0, -1], [-8, 2, 2, 2, 2], [-5, 3, 0, 0, 5], [1, 1, 0, -1, 0], [3, 5, 0, 2, 0], [3, 2, -1, 1, -1], [-2, -1, 3, -1, -1], [-4, 1, -2, -1, -1], [-6, 4, 0, -2, 4], [-5, 2, 1, -7, -3], [5, 6, -1, 0, -3], [7, -4, 3, 3, 2], [5, 0, 1, -1, 2], [-3, -1, -2, 2, 3], [6, -2, 0, 2, -2], [-3, -5, -4, 4, 1], [10, 7, -2, 1, 3], [-2, 3, -3, 3, 4], [-5, 2, -4, -7, -6], [-5, 3, -4, -2, -5], [-1, -6, -1, -1, -4], [0, 1, 3, 1, 4], [4, 2, 2, 0, -4], [5, -3, 2, 0, 3], [-9, 1, 3, 4, 5], [1, -6, 1, 0, 1], [2, -7, 1, -1, -5], [1, 0, -3, -3, -2], [-3, 1, -2, 6, 3], [6, -7, 1, -1, 0], [6, -3, -3, 1, -1], [-4, 9, 5, -3, 1], [2, 2, 4, 2, 4], [-3, 13, 2, -2, 5], [7, 0, 1, 7, 1], [-1, -3, 0, -4, -7], [-10, -7, -1, 1, -3], [17, -3, 6, 0, 3], [1, 6, 3, 3, 1], [11, 1, 2, 4, 3], [-8, 4, 0, 4, 0], [-3, -1, -2, 2, 1], [14, -2, 2, 4, -4], [2, -8, 1, 0, -8], [7, 0, -2, 3, -2], [-1, 7, 2, 8, 5], [-12, 4, 4, 4, 4], [4, 1, 5, 1, 6], [20, -4, 2, 4, 0], [3, -11, 4, 4, 1], [10, -6, -6, 2, 0], [-2, 11, -3, 0, 2], [-6, 7, -11, 1, 1], [21, 1, 3, -2, 1], [-8, -7, 5, -1, -4], [-11, 8, 3, -9, -1], [0, -14, 6, 3, 3], [1, -6, 3, 3, -2], [2, 3, -1, -2, 0], [-15, 11, -6, -4, -3], [4, -1, -7, -7, -9], [-2, 6, -2, -6, 4], [-1, -3, 0, -4, -1], [1, -12, -1, 3, 4], [9, 6, -3, 9, 5], [3, 3, -4, -5, 2], [-8, 1, -7, 1, -2], [7, -3, -2, -6, -11], [-3, -1, -2, 2, 1], [4, 3, 5, 9, 7], [-17, 5, -4, -6, -7], [9, 5, 6, -2, 2], [-2, 3, 2, 3, 1], [-2, -8, 8, -2, 1], [-4, 2, -8, 2, 4], [22, 0, 2, 6, 5], [-6, -2, 2, -1, 1], [7, -9, -4, -1, -6], [-19, 1, -6, -8, -7], [-11, 1, -1, -2, -3], [1, -1, -6, 8, -1], [-2, -1, 1, -5, 1], [5, 2, 5, 4, 9], [1, 7, -5, -4, -9], [2, -10, 10, -4, -8], [11, -12, 5, 3, -3], [1, 2, 10, -1, 6], [2, -1, -7, 6, 4], [-12, 7, 11, 0, 0], [10, -2, -3, -2, -12], [3, 5, 0, -4, 3], [23, -5, 8, 4, -4], [-19, 4, 0, -5, 6], [2, 13, -7, 1, -1], [-25, -2, 1, -8, -1], [15, 3, -2, -4, 1], [-8, -2, 2, -11, -3], [1, -11, 10, 6, 5], [-21, 14, 7, -3, 0], [-1, -1, -8, 8, -1], [0, -8, 14, -8, -8], [25, -5, -2, -2, -9], [9, -8, 1, 5, 4], [3, -1, -4, -2, -7], [20, -2, -2, 1, -7], [7, -17, 14, 7, 10], [-8, -2, -4, -8, -12], [17, 1, 6, 2, -1], [-8, -16, 3, 2, 8], [15, -5, 4, -8, -11], [-2, -12, -8, 9, 7], [0, -14, -7, -2, -10], [-27, 11, -4, -10, 3], [0, -2, -8, -2, -8], [0, -12, -2, 4, 2], [8, 0, -6, 0, -12], [1, 2, -4, -5, -8], [4, 14, -2, 2, 2], [1, 5, -2, 2, -3], [9, -8, 3, -3, -6], [-3, 1, -2, 10, 7], [-5, 14, -6, -9, -4], [-4, 13, 1, 7, 5], [1, -9, -2, 0, -1], [23, 7, 6, -8, -1], [12, -15, -9, 3, -6], [6, -12, 0, 0, -4], [-20, 5, -7, -11, 1], [-19, 10, -1, -5, 8], [-3, -15, 6, -2, -13], [-14, 4, 0, -8, 3], [-22, 12, -4, -12, -8], [13, 0, 1, 3, -8], [-2, 3, 7, 4, -2], [-20, 8, 2, 5, 5], [10, -19, 4, 5, 3], [-12, 9, 3, -3, 5], [0, -12, 4, -2, -7], [27, -8, -1, 10, 1], [-31, 0, -5, 5, -4], [-6, 0, 0, -2, 11], [-28, 14, -2, -7, 5], [20, -10, 12, 8, 10], [3, 28, -2, -1, 4], [22, 5, -13, 1, 1], [15, -7, -2, -8, -19], [5, -13, -4, 8, -5], [4, -4, -4, 2, 5]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_287_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_287_2_a_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_287_2_a_e(:prec:=5) chi := MakeCharacter_287_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_287_2_a_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_287_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_287_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![3, 6, -4, -6, 1, 1]>,<3,R![-1, -3, 10, 0, -4, 1]>],Snew); return Vf; end function;